# theory of complexity – LTL and alternation

Apparently, the verification of the LTL model on Kripke structures is PSPACE-complete (see this question). Usually, when something in finite model theory is difficult to apply with PSPACE, there is a simple proof using the APTIME incarnation of PSPACE and alternation in logic. Or there is a reduction of QBF (the evaluation problem for quantified Boolean formulas).

In the case of LTL, I find it difficult to see the slightest link with the alternation. In fact, reducing the number of documents cited in the associated question can be offset by a single G operator and other Boolean connectors and (many) X operators. This makes sense because it reduces determinist Turing machines (they use the PSPACE incarnation of PSPACE), not the alternates. However, this implies that an LTL fragment that exhibits no apparent alternation presents the same complexity.

My question is this: is there an intuition between LTL and alternation? In particular, an intuition that remains valid for fragments as above which, syntactically, do not seem to allow alternation.